How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? Let $X$ be a separable and locally compact metric space and let $P\colon X\times \mathcal{B}(X) \rightarrow \mathbb{R}$ be the transition probabability kernel of a homogeneus Markov chain on $X$. Define $$Pf(x)=\int_{X} f(y)\,P(x,dy)\;\;\; (f\in C(X), x\in X),$$
where $C(X)$ is the space of bounded continuous functions $f:X\rightarrow \mathbb{R}$. We say that $P$ has the Feller property if $Pf \in C(X)$ for all $f\in C(X)$.
My question is: how to prove the following theorem (Proposition 6.4.2, Meyn and Tweedie, Markov chains and stochastic stability: http://probability.ca/MT/Chap6.pdf):

Assume that $P$ has the Feller property, and that there exists a unique probility measure $\pi$ on $\mathcal{B}(X)$ such that for every $x\in X$: $$P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi\;\;\;(n\to\infty).$$
  Then the family $\{P^n f: n\in\mathbb{N}\}$ is equicontinuous on compact subsets of $X$ whenever $f\in C(X)$ ($\stackrel{w}{\rightarrow}$ denotes the weak convergence).

The authors claim that it follows directly from Ascoli's Theorem. However I can not see this. In order to be able to apply Ascoli's theorem we need to know that for each compact set $C$ at least some subseqence $(P^{k_n} f)_{n}$ is uniformly convergent on $C$. Since $C$ is compact we may choose $x_n \in C$ such that $$|P^n f(x_n)-\pi(f)|=\max_{x\in C} |P^n f(x)-\pi(f)| \;\;\;(n\in\mathbb{N}),$$ where $\pi(f)=\int_X f(y)\,\pi(dy)$. So it suffices to show that there exists a strictly increasing sequence $(k_n)_{n\in\mathbb{N}}$ of natural numbers such that $$P^{k_n}f(x_{k_n})\to \pi(f)$$ but I have a problem in this place. I would be grateful for any ideas.
 A: This doesn't answer the question, but the result does hold if we assume the  strong Feller property ($Pg \in C(X)$ for every bounded measurable $g$).
Namely, suppose $f \in C(X)$.  Then 
$$\begin{align*}|P^{n+1} f - \pi f| &= |P(P^n f - \pi f)|\\
&\le P |P^n f - \pi f| \\
&\le P \sup_{m \ge n} |P^m f - \pi f|.\end{align*}$$
Set $F_n = \sup_{m \ge n} |P^m f - \pi f|$.  By construction, $F_n$ is monotone decreasing, so by the monotonicity of $P$, $P F_n$ is also monotone decreasing.  Moreover, $\|F_n\|_\infty \le 2 \|f\|_\infty$.  And since $P^m f \to \pi f$ pointwise, we have $F_n \to 0$ pointwise.  So by monotone convergence, we also have $P F_n \to 0$ pointwise.  Finally, by the strong Feller property, each $P F_n$ is continuous.  So $P F_n$ is a monotone decreasing sequence of continuous functions which converges to 0 pointwise.  By Dini's theorem, we have $P F_n \to 0$ uniformly on compact sets.  This gives us that $P^{n+1} f \to \pi f$ uniformly on compact sets; in particular, $\{P^n f\}$ is equicontinuous on compact sets.
I got this argument from the proof of Proposition 2.3 of this paper by Schilling and Wang.
